N2012 P2 Q9 [Modified]

In an opinion poll before an election, a sample of \(30\) voters is obtained. The number of voters in the sample who support the Alliance Party is denoted by \(A\). State in context, two conditions that must be assumed for \(A\) to be modelled by a Binomial Distribution.

1. It is given that \(5\%\) of the voters in the election support the Alliance Party.
   1. What is the probability that fewer than \(6\) of the \(30\) voters support the Alliance Party?[2]
   2. What will be the most likely number of voters who support the Alliance Party in a sample of \(40\) voters?[2]
   3. It is given that, in a sample of \(50\) voters, the probability that at least \(m\) voters who support the Alliance Party is less than \(0.4\). Find the least possible value of \(m\).[3]
   4. If a sample of \(n\) number of voters is obtained and the probability of getting at least a supporter is not more than \(0.75\). Find the maximum value of \(n\).[3]
2. In the next election, another sample of \(30\) voters is obtained and the percentage of the voters is \(p\) where \(p>50\). It is given that the probability of getting \(15\)to \(20\)supporters is \(0.3\). Find the value of \(p\).[4]